The present invention relates to the art of medical diagnostic imaging. It finds particular application in conjunction with magnetic resonance imaging (MRI) and will be described with particular reference thereto. It is to be appreciated, however, that the invention will also find application in other imaging applications in which data is transformed between frequency and time or spatial domains.
One of the pursuits in MRI is increased imaging speed while maintaining or only minimally degrading the spatial resolution of a resultant image. In this spirit, fast imaging sequence such as echo-planar imaging (EPI) interleaved EPI, fast spin echo (FSE) as well as other imaging sequences are employed. In using such fast sequences, MRI systems are pushed to and beyond their hardware limits. Thus, to achieve increased imaging speed, imaging sequences are constantly being optimized. Along with this optimization, the process of reconstructing data obtained by such imaging sequences must be optimized.
Heretofore, magnetic resonance imaging subjects have been positioned in a temporally constant magnetic field such that selected dipoles preferentially align with the magnetic field. A radio frequency pulse is applied to cause the preferentially aligned dipoles to resonate and emit magnetic resonance signals of a characteristic resonance radio frequency. The radio frequency magnetic resonance signals from the resonating dipoles are read out for reconstruction into an image representation.
In a two-dimensional Fourier transform imaging technique, a read gradient is applied during the read out of the echo for frequency encoding along a read axis and a phase-encode gradient is pulsed to step phase-encoding along a phase-encode axis between echoes. In this manner, each echo generates a data line in k-space. The relative phase-encoding of the data lines controls their relative position in k-space. Conventionally, the data line with zero phase-encoding extends across the center of k-space. Data lines with a phase-encoding gradient stepped in progressively positive steps are generally depicted as being above the center line of k-space; and, data lines with progressively negative phase-encoding steps are depicted as being below the center line of k-space. In this manner, a matrix, such as a 256.times.256 or a 512.times.512, etc., matrix of data values in k-space is generated. Fourier transformation of these values generates a conventional magnetic resonance image.
To strengthen the received magnetic resonance signals, the initial signal is commonly refocused into an echo. This may be done by reversing the polarity of a magnetic field gradient to induce a field or gradient echo. Analogously, the radio frequency excitation pulse may be followed with a 180.degree. pulse to refocus the signal as a spin echo. Moreover, by repeating the reversing of the magnetic field gradient, a series of gradient echoes can be generated following each radio frequency excitation pulse. Analogously, a series of spin echoes can be generated following each radio frequency excitation pulse by repeating the 180.degree. radio frequency refocusing pulse. As yet another option, a single radio frequency excitation pulse can be followed by a mixture of spin and gradient echoes. See, for example U.S. Pat. No. 4,833,408 of Holland, et al.
In a single shot echo planar imaging (EPI) sequence, a single radio frequency excitation pulse or shot of arbitrary tip angle can be followed by a sufficient number of gradient reversals to generate an entire set of data lines. The magnetic resonance data from the object is collected during a series of echoes with an oscillatory read gradient that encodes the image object in the direction of the field gradient. See, e.g., P. Mansfield, J. Phys. Chemistry, Vol. 10, pp. L55-L58 (1977). In addition, a series of phase-encoding gradient pulses orthogonal to the read gradient direction are applied before each echo to step the data lines through k-space. The image of the object is preferably obtained with two one-dimensional inverse Fourier transforms of the echo data. This single shot EPI technique offers an ultra fast imaging technique for dynamic imaging in a sub-second time scale.
Multi-shot EPI techniques offer improved image quality over single-shot EPI techniques. In multi-shot EPI imaging, phase-frequency space or k-space is divided into a plurality of segments, e.g., 3 to 16 segments. After resonance excitation, the read gradient is oscillated to generate a train of echoes, hence data lines, in each of the segments of k-space. After another excitation, a different one of the data lines in each segment is generated. This process is repeated until k-space is filled in this interleaved fashion.
The one-dimensional inverse Fourier transform most often used to reconstruct the data obtained during an imaging sequence is the "fast Fourier transform," also known as the Cooley-Tukey algorithm. Although magnetic resonance literature often refers to a "Fourier transform", those skilled in the art understand that a fast inverse Fourier transform is being used. The universal use of the fast Fourier transform is evidenced by image sizes that are integer powers of two, such as 512.times.512, 256.times.256, etc. The use of square image matrices is dictated by the integer power of 2 requirements of the fast Fourier transform algorithm. This transform performs a Fourier transform operation on an N by N matrix with only Nlog.sub.2 N mathematical operations. The fast Fourier transform algorithms are limited because N was required to be an integer power of an integer known as the Radix value, most commonly 2. The dramatic increase in speed was considered more than worth the limitation of the length of the data lines. Note that for a data line with 512 samples (N=512), the discrete Fourier transform required over 260,000 mathematical operations; whereas, the fast Fourier transform only requires about 4,600. Because computing time is roughly proportional to the number of mathematical operations, the discrete Fourier transform required about 56 times as long as a fast Fourier transform to process a 512 sample line. Due to the exponent in this relationship, larger data lines achieved an even more dramatic time savings. The fast Fourier transform reduced the computing time sufficiently that fast Fourier transforms became a standard computer subroutine.
Current MRI systems include hardware constraints such as time-varying gradient wave forms, non-linear magnetic gradients and rate of data sampling. Time-varying gradient wave forms or non-linear gradients can cause non-equidistant k-space data. For example, the "rise time" of a magnetic gradient pulse generator produces gradient pulses that are trapezoidally shaped rather than rectangularly shaped. In other words, the gradient pulses have "ramp up" and "ramp down" at their leading and trailing edges. Thus, the leading and trailing edges of the gradient pulses change with respect to time. Typically, data sampled during these ramped edges are also non-equidistant in k-space.
In other applications, time varying gradient pulses, such as sinusoidal gradient profiles, are intended. When data sampling during such time-varying gradient profiles, non-equidistant k-space data is obtained. This non-equidistant k-space data must be corrected before Fourier-transformation.
To reconstruct an image from non-equidistant k-space data, the data is first "gridded" to generate a set of uniformly spaced k-space data. Subsequently, the data is inverse fourier transformed to generate an image representation or image matrix.
In reconstructing an image from EPI data obtained with a time varying read-out gradient, generally a well known "gridding" algorithm is used to map the resultant non-uniform k-space data to uniformly spaced data. See Jackson, et al., IEEE Trans. Med. Imaging 10,473(1991). The final image is then obtained by Fourier transforming the data.
One of the problems of this gridding method stems from the use of a kernel function for convolution in interpolating the non-equidistant k-space data to an equidistant data set. The kernel function introduces errors as well as blurring into the final image.
As an alternative gridding method for Fourier transform based image reconstruction, a least squares estimator matrix is used to directly convert non-equidistant k-space data through a matrix multiplication into a uniformly spaced image matrix. See U.S. Pat. No. 4,982,162 to Zakhor et. al.
To efficiently generate an image matrix from non-equidistant k-space data with minimal loss of spatial resolution, reconstruction methods other than inverse fourier transforms may be used. One such reconstruction method is the linear system solution or singular value decomposition (SVD) method.
The Zakhor method suffers in that weighing factors for different linear equations are not considered in the least squares error minimization. Further, the image matrix size is fixed by the number of data samples. Still further, the method is not flexible for other applications such as zero padding.
The present invention provides a new and improved apparatus and method for generating improved quality images from data sets or matrices of non-equidistant k-space data.